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Deep Curve-dependent PDEs for affine rough volatility

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Deep Curve-dependent PDEs for affine rough volatility

Antoine Jacquier
Mugad Oumgari
Published on 6/6/2019
Options
Derivatives
Machine learning
Deep learning
AI

This paper introduces a novel deep learning-based algorithm for pricing European options in affine rough stochastic volatility models, such as the rough Heston and rough Bergomi models. The key insight is that, under specific affine assumptions, the option price depends only on forward-looking curves (e.g., forward variance) rather than the entire path of the volatility process. This reduces the pricing problem to a curve-dependent PDE (CPDE), which is a special case of path-dependent PDEs. The authors develop a backward discretization scheme for the CPDE and solve it using a deep learning approach inspired by the deep BSDE method of E, Han, and Jentzen. The algorithm is validated on the rough Heston model, showing accurate pricing compared to Monte Carlo simulations with lower computational cost. The paper also establishes market completeness and no-arbitrage conditions for the considered model class. This work provides a promising alternative to Monte Carlo methods for pricing in rough volatility models, with potential extensions to local-stochastic volatility and path-dependent options.

Highlights

  • 1Introduces a deep learning-based algorithm for pricing options in affine rough stochastic volatility models by solving curve-dependent PDEs (CPDEs).
  • 2Shows that for European vanilla options, the pricing function depends on forward-looking curves rather than the whole path, reducing complexity.
  • 3Proves market incompleteness and absence of arbitrage for rough local stochastic volatility models under specific assumptions.
  • 4Develops a backward discretization scheme combined with deep learning to solve CPDEs numerically, avoiding Monte Carlo simulations.
  • 5Validates the algorithm on the rough Heston model, demonstrating its accuracy and efficiency compared to Monte Carlo methods.

Methods

  • M
    Curve-dependent PDE (CPDE) formulation derived from the Feynman-Kac theorem for rough volatility models.
  • M
    Deep learning-based backward algorithm inspired by E, Han, and Jentzen (2017) for solving high-dimensional PDEs.
  • M
    Discretization of the pricing CPDE using a backward scheme with neural network approximations.
  • M
    Monte Carlo simulation with the Hybrid scheme for benchmarking and comparison.

Results

  • R
    The pricing function for European options in affine rough volatility models depends only on forward curves, not the full path.
  • R
    The deep CPDE algorithm accurately prices options in the rough Heston model, with errors comparable to Monte Carlo but with lower computational cost.
  • R
    The method handles non-Markovian dynamics effectively, providing a viable alternative to simulation-based approaches.
  • R
    Numerical experiments show convergence and stability of the deep learning scheme for various model parameters.
  • R
    The algorithm can be extended to path-dependent options, though the full path is then required.
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